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I was playing with some graphs and I noticed that $\frac{1}{x}-\frac{1}{x+\pi}+\frac{1}{x+2\pi}$ looked similar to $\csc x$.

I then plugged in (to desmos.com) $\sum_{n=-500}^{500}\frac{(-1)^n}{x+nπ}$ and found that it was pretty much $\csc{x}$ on the interval near $(-500π,500π)$.

So, my question is, why is it that $\sum_{-\infty}^\infty \frac{(-1)^n}{x+nπ} = \csc x$

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    sorry, fixed it @Rohan – 2017-02-14
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    http://math.stackexchange.com/questions/1963263/prove-cscx-sum-k-infty-infty-frac-1kxk-pi – 2017-02-14

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